Problem: The sum of two numbers is $52$, and their difference is $26$. What are the two numbers?
Solution: Let $x$ be the first number, and let $y$ be the second number. The system of equations is: ${x+y = 52}$ ${x-y = 26}$ Solve for $x$ and $y$ using elimination. Add the top and bottom equations together. $ 2x = 78 $ $ x = \dfrac{78}{2} $ ${x = 39}$ Now that you know ${x = 39}$ , plug it back into $ {x+y = 52}$ to find $y$ ${(39)}{ + y = 52}$ ${y = 13}$ You can also plug ${x = 39}$ into $ {x-y = 26}$ and get the same answer for $y$ ${(39)}{ - y = 26}$ ${y = 13}$ Therefore, the larger number is $39$, and the smaller number is $13$.